Vol. 26, No. 3, 1968

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Bases in Hilbert space

William Jay Davis

Vol. 26 (1968), No. 3, 441–445

A sequence (xi) of elements of a Hilbert space, , is a basis for if every h ∈ℋ has a unique, norm-convergent expansion of the form h = aixi, where (ai) is a sequence of scalars. The sequence is minimal if there exists a sequence (yi) ⊂ℋ such that (xi,yj) = δij. Every basis is minimal, and the sequence (ai) in the expansion of h (above) is given by ai = (h,yi). In this paper, we restrict our attention to real Hilbert space.

We derive, from classical characterizations of bases in B-spaces, criterea for (xi) to be a basis for , as well as for (xi) to be minimal in . We show that the sequence is minimal if and only if there are sequences (gi) ⊂ℋ whose Gram matrices have a prescribed form. Similar conditions are obtained for (xi) to be a basis for .

Mathematical Subject Classification
Primary: 46.15
Received: 13 December 1966
Published: 1 September 1968
William Jay Davis